PHYSICAL REVIEW B 84, 125319 (2011)
Valley degeneracy in biaxially strained aluminum arsenide quantum wells
S. Prabhu-Gaunkar,1 S. Birner,2 S. Dasgupta,2 C. Knaak,2 and M. Grayson1 1Electrical Engineering and Computer Science, Northwestern University, Evanston, Illinois 60208, USA 2Walter Schottky Institut, Technische Universitat¨ Munchen,¨ Garching DE-85748, Germany (Received 13 September 2010; revised manuscript received 19 May 2011; published 16 September 2011) This paper describes a complete analytical formalism for calculating electron subband energy and degeneracy in strained multivalley quantum wells grown along any orientation with explicit results for AlAs quantum wells (QWs). In analogy to the spin index, the valley degree of freedom is justified as a pseudospin index due to the vanishing intervalley exchange integral. A standardized coordinate transformation matrix is defined to transform between the conventional-cubic-cell basis and the QW transport basis whereby effective mass tensors, valley vectors, strain matrices, anisotropic strain ratios, piezoelectric fields, and scattering vectors are all defined in their respective bases. The specific cases of (001)-, (110)-, and (111)-oriented aluminum arsenide (AlAs) QWs are examined, as is the unconventional (411) facet, which is of particular importance in AlAs literature. Calculations of electron confinement and strain for the (001), (110), and (411) facets determine the critical well width for crossover from double- to single-valley degeneracy in each system. The biaxial Poisson ratio is calculated for the high-symmetry lower Miller index (001)-, (110)-, and (111)-oriented QWs. An additional shear-strain component arises in the higher Miller index (411)-oriented QWs and we define and solve for a shear-to-biaxial strain ratio. The notation is generalized to address non-Miller-indexed planes so that miscut substrates can also be treated, and the treatment can be adapted to other multivalley biaxially strained systems. To help classify anisotropic intervalley scattering, a valley scattering primitive unit cell is defined in momentum space, which allows one to distinguish purely in-plane momentum scattering events from those that require an out-of-plane momentum component.
DOI: 10.1103/PhysRevB.84.125319 PACS number(s): 73.21.Fg, 73.50.Bk
I. INTRODUCTION Various formalisms have been developed to understand valley occupancy in a multivalley system. AlAs QWs are Calculating valley degeneracy in a quantum well (QW) slightly strained with respect to the GaAs substrate, so one requires a comprehensive treatment of strain, quantum confine- must consider quantum confinement, strain, and piezoelectric ment, and piezoelectric fields since all contribute at compara- fields to estimate the electron subband energy and degeneracy ble energy scales. Of the various multivalley semiconductors, the indirect-bandgap zinc blende semiconductor aluminum of the different electron valleys. Stern and Howard modeled arsenide (AlAs) with its bulk threefold valley degeneracy is two-dimensional (2D) confinement of electron valleys in Si for arbitrary crystal plane orientations neglecting strain of particular interest (Fig. 1) because its heavy anisotropic 21 electron mass allows for large interaction effects,1 and its effects. Van de Walle theoretically studied the absolute energy level for an unstrained semiconductor heterojunction, near-perfect lattice match to GaAs substrates allows for 22 high-mobility, modulation-doped quantum wells (QWs).2,3 considering strain effects only for bulk systems. Smith AlAs/AlGaAs QWs can reach mobilities of the order of μ = et al. calculated the strain tensor theoretically for the case of 23 100 000 cm2/Vs in (001)-facet QWs4 and in the high-mobility (001)- and (111)-oriented QW superlattices. Although Caridi direction of anisotropic (110)-facet QWs.5 Unconventional and Stark derived the complete strain tensor for arbitrarily facets such as (411) have also proven useful in identifying oriented substrates with cubic symmetry, they ignored a exchange effects like quantum Hall ferromagnetism in AlAs critical shear component, which arises in the high Miller 24 QWs.6,7 Evidence for a QW width crossover from double- index directions under relaxation. De Caro et al. calculated to single-valley occupation has been shown for (001)AlAs the shear strain with commensurability constraint equations wells,8,9 as has evidence for single-valley occupancy in wide accurately only for the low index (100), (110), and (111) (110)AlAs wells.5 Dynamic control of the valley degeneracy has been realized with uniaxially strained (001)AlAs QWs to induce valley degeneracy splitting.1,10,11 Such studies can quantify interaction effects, calibrating valley strain suscepti- bility and valley effective mass.10,12–14 Quantum confinement to a one-dimensional multivalley system has been achieved in cleaved-edge overgrown quantum wires15,16 and quantum point contacts.17 Novel interaction effects in QWs include anisotropic composite Fermion mass18 and valley skyrmions, whereby electrons populate linear superpositions of two FIG. 1. (Color online) (Left) Real-space depiction of AlAs QW 19 valleys at once. Such interaction effects that result from structures defining the transport basis axes: a,b, and c. (Right) 20 exchange splitting of a perfect SU(2) symmetry may prove Brillouin zone for bulk AlAs defining the conventional-cubic-cell useful in future quantum device applications, where the valley basis x,y, and z. Note the three degenerate valleys that are occupied degree of freedom functions as a pseudospin. with electrons as indicated by ellipsoidal equienergy contours.
1098-0121/2011/84(12)/125319(15)125319-1 ©2011 American Physical Society S. PRABHU-GAUNKAR et al. PHYSICAL REVIEW B 84, 125319 (2011) orientations.25 Yang et al. corrected the constraint equations strain tensor, degeneracies, piezoelectric fields, and valley- and calculated piezoelectric fields arising from the shear splitting energies for the specific cases of (001)-, (110)-, strain components for the general case of a pseudomorphic (111)- as well as the unconventional (411)-oriented QWs film.26 Adachi calculated the strain tensor theoretically for illustrate how valley degeneracies can be engineered. We the case of bulk and superlattice structures but did not review the crossover width calculation for double-to-single address the case of single QW structures.27 Hammerschmidt valley occupation in the (001)-, (110)-, and (411)-facet QWs. et al. focus on the case of isotropic planar strain arising in For the high-symmetry low-Miller-index orientations (001), QWs but do not carry out a combined treatment of strain (110), and (111), we calculate the biaxial Poisson ratio, and for energy along with quantum confinement energy and how these the low-symmetry high-Miller-index (411) facet, there arises affect valley degeneracies.28 Theoretically, Rasolt proposed a shear component in the strain tensor and, consequently, a an understanding of the various continuous symmetries and shear-to-biaxial strain ratio. We calculate intervalley scattering symmetry breaking in a multivalley system in terms of an ratios for valley-degenerate AlAs QWs on various facets SU(N) symmetry, where N is the valley degeneracy number,20 in Sec. VIII and refer to the valley-scattering unit cell for but did not explicitly derive the vanishing intervalley exchange intuition. We complete our analysis by comparing the standard integral, which underlies this theory. Material-specific calcu- 2D Bravais lattice valley representation to our valley cell lations of quantum confinement and strain in high-mobility representation. To generalize the formalism for calculating valley subband energies in arbitrarily oriented facets, we end Si and Ge structures are well studied for high-symmetry 43 facets29–40 and, recently, such calculations have been been in Sec. IX by adapting this notation for miscut samples. made in other quantum confined systems as well.41,42 However, a combined treatment of quantum confinement, strain, and II. VALLEY EXCHANGE INTEGRAL AND VALLEY INDEX piezoelectric fields to determine valley degeneracy in QWs is AS A PSEUDOSPIN lacking, especially, for low-symmetry facets. Just as electrons with different spins are distinguishable and To eventually model transport in such valley-degenerate have no exchange term in their interaction energy, electrons in systems, one must also consider the extra scattering channel different valleys can be shown to be effectively distinguishable not present in single-valley systems, namely, intervalley from each other due to the vanishingly small intervalley scattering. It is therefore useful to introduce a k-space unit cell exchange energy. The valley index can thus be treated as that permits visualization of momentum-scattering events in a a pseudospin index. Rasolt20 discusses symmetry breaking geometry that is natural to the quantum-confinement direction. in a multivalley system but does not explicitly derive this However, the standard depiction of a 2D Brillouin zone of exchange integral. Previous work on Si quantum dots to study a quantum-confined valley-degenerate system21 projects all Kondo effect44–46 calculates the exchange integral only for the valleys to a single 2D plane. Such a depiction loses information special case of identical spatial wavefunctions on Si quantum about the out-of-plane momentum-scattering component that dots. For completeness, in this section we explicitly derive was projected out, which is necessary to determine the full the inter and intravalley exchange integrals for arbitrary wave momentum-scattering matrix element. Thus it is useful to functions to justify the valley index as a pseudospin, with develop a graphical representation for the unit cell in k space exchange interaction proportional to δ , where τ and τ are that can clearly elucidate how valleys are coupled with both ττ valley indices. in-plane and out-of-plane momentum-scattering events. An electron wave packet within a single valley is described The paper is organized as follows. In Sec. II, the exchange with a weighted integral over Bloch functions. The normalized energy calculations demonstrate why the valley index can wave packet ψ in valley τ is chosen to be in a spin-polarized function as a pseudospin index. In Sec. III, we develop the τ,σ state σ described by the spinor χσ : notation and formalism for determining valley degeneracy in multivalley strained semiconductor QWs. The three subband 1 τ = √ 3 i(q +k)·r energy components are defined: kinetic energy, confinement ψτ,σ (r) d kAke uqτ +k(r)χσ , (1) V energy, and strain energy. The two useful coordinate bases are defined as well: the conventional-cubic-cell basis and the where uqτ +k(r) is the component of the Bloch function periodic transport basis, as well as the coordinate transformation matrix in the Bravais lattice, Ak is the complex amplitude for a that transforms between them. Single-electron analytical solu- particular k, qτ + k is the total crystal momentum. The center tions are provided to allow easy identification of characteristic of the valley is denoted with qτ and k denotes the additional energy scales for multivalley systems. The explicit criteria small momentum deviation away from this valley center, for valley degeneracy in arbitrarily oriented biaxially strained and V is the volume of the system. Assuming k is much QWs are defined in Sec. IV. Section V describes how shear smaller than the Umklapp vector for the lattice and taking the ∼ strain can induce piezoelectric fields in the QW. In Sec. VI,we envelope-function approximation uqτ +k = uqτ , we obtain introduce the valley-scattering unit cell; a three-dimensional (3D) primitive unit cell with the same volume as the Brillouin 1 iqτ ·r 3 ik·r ψτ,σ (r) = √ uqτ (r)e χσ d kAke zone but with the full symmetry of the reciprocal lattice vectors V that lie in the Miller index plane. 1 τ = √ iq ·r Momentum-space illustrations provide intuition for vi- uqτ (r)e χσ φ(r), (2) V sualizing intervalley scattering and valley degeneracies. Section VII applies the developed formalism to the case of where φ(r) is the slowly varying envelope function. The AlAs QWs, whereby the projected in-plane transport masses, Coulomb exchange energy integral between the two electron
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wave functions ψτ,σ (r) and ψτ,σ (r) in valleys τ and τ with energy shift caused by lattice mismatch of the QW with respect coordinates r1 and r2 is given by to the substrate. Note that the Miller index M is not explicitly superscripted because it is common to all valleys. To calculate −e2 Eτ,τ,σ,σ = dr dr ψ (r )ψ∗ (r ) these terms in Eq. (7), we need to find the in-plane (parallel to ex 1 2 |r − r | τ,σ 1 τ,σ 2 σ σ 1 2 the QW) and out-of-plane (confinement direction) components ∗ of the inverse mass tensor of the corresponding electron valley × ψτ,σ (r2)ψτ,σ (r1). (3) as well as the various strain-tensor components in the QW. Substituting Eq. (2)inEq.(3), we can show that two par- We start by introducing two useful bases, the conventional- ticles with different valley indices behave like distinguishable cubic-cell (CCC) basis x = (x,y,z) and the transport basis particles by virtue of their vanishing exchange integral, a = (a,b,c), along with the coordinate transformation matrix RM , which transforms between them, δ ∗ Eτ,τ,σ,σ = σ σ dr dr φ(r )φ∗(r )φ(r )φ (r ) ex V 2 1 2 1 2 2 1 a = RM x. (8) 2 −e ττ iQ ·(r2−r1) The CCC basis has the x, y, and z axes aligned along the axes × e uqτ |r1 − r2| of the cubic cell of the reciprocal lattice of the crystal, see × ∗ ∗ Fig. 1 (right). The transport basis is chosen with the ab plane (r1)uqτ (r2)uqτ (r2)uqτ (r1)(4) parallel to the QW and the c direction perpendicular to this with intervalley scattering wave vector plane, see Fig. 1 (left). Thus if the Miller index of the growth plane is (hkl), the perpendicular unit vector in the transport Qττ = qτ − qτ , (5) basis is c = (h,k,l) √ 1 . To uniquely define the a and b h2+k2+l2 † = where the inner product of the spinors is given by χσ χσ δσ σ . directions in the transport basis, we take a to be the in-plane This delta function denotes that the exchange integral vanishes unit vector with the lowest Miller index and b the unit vector when the two spin indices are different σ = σ. Analogously, that maintains a right-handed coordinate system a × b = c. if τ = τ, then Qττ is of the order of an Umklapp vector, With this definition we obtain a unique set of axes. These also and the rapidly oscillating complex exponential makes the define the components of the coordinate transformation matrix integral vanish for envelope functions larger than a few lattice RM , whereby a, b, and c are the top, middle, and bottom rows constants. On the other hand, if τ = τ, then Qττ = 0, the expo- of the coordinate transformation matrix RM , respectively. In nential term is unity, and the integral remains finite. Therefore, what follows, vectors and tensors expressed in the x-basis are the dependence of the exchange integral (4) on the valley index unprimed and in the a-basis are primed. can be approximated with a second delta function to notate the The mass tensor is naturally expressed in the unprimed vanishing exchange integral between different valleys: CCC x-basis, so by transforming it to the primed transport ∗ a-basis, we can easily extract the in-plane and out-of-plane τ,τ,σ,σ δσ σ δττ 2 E = dr φ(r )φ (r )|u τ (r )| confinement masses for a given electron valley. The matrix ex V 2 1 1 1 q 1 inverse of the mass tensor for the τ-valley in the x-basis is − 2 τ τ −1 ∗ e 2 denoted by w = (m ) . We assume parabolic bands so that × dr2φ (r2)φ(r2) |uqτ (r2)| . (6) |r1 − r2| the mass is independent of the wave vector k. Applying the transformation (wτ ) = RM (wτ )(RM )−1, we obtain the inverse We note that Eq. (6) cannot be simplified because the mass tensor of the τ-valley in the primed transport a-basis: Coulomb potential can vary rapidly over small distances of ⎡ ⎤ τ τ τ order a lattice constant. waa wab wac We conclude by virtue of this vanishing intervalley ex- ⎢ ⎥ (w τ ) = ⎣w τ w τ w τ ⎦ . (9) change interaction that wave functions in different valleys can ba bb bc τ τ τ be treated like distinguishable particles and the valley index is wca wcb wcc 20 a valid pseudospin index. As was pointed out by Rasolt, the τ The upper left 2 × 2 submatrix w × gives the inverse in-plane valley pseudospin constitutes an SU(N) group, where N is the 2 2 mass tensor and w τ gives the inverse of the confinement mass number of electron valleys. cc perpendicular to the QW.
III. VALLEY SUBBAND ENERGY A. Quantum confinement energy Crystal symmetry dictates that multiple energy-degenerate We can now define the first energetic term in Eq. (7), valleys will occur whenever a local conduction band minimum τ the quantum confinement energy E0 (k), using the coordinate exists away from the origin in momentum space (the point). transformation matrix and out-of-plane component of the When such a multivalley system is quantum confined in a layer inverse mass tensor in the primed a-basis. This is calculated = τ with planar Miller indices M (hkl), the energy E (k)inthe from the ground-state-energy solution of the 1D Schrodinger¨ lowest subband in valley τ is equation for a particle confined along c direction with in-plane τ = τ + τ + τ momentum k, E (k) E0 (k) T (k) E , (7) − 2 h¯ d τ dψk(c) τ where k is the 2D in-plane momentum relative to the τ-valley w (c) + V (c,k)ψk(c)=E (k)ψk(c), τ τ 2 dc cc dc 0 minimum, E0 (k) is the ground confinement energy, T (k) is the in-plane kinetic energy, and Eτ is the strain-induced (10)
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τ where w cc(c) is the diagonal component of the reciprocal C. Strain energy mass tensor in the a-basis. Due to the different reciprocal τ τ We can now define the final term of Eq. (7), the strain masses in the barrier (wB ) and the well (wW) layers, wave- τ 47 energy E , in terms of the strain tensor and the coordinate function derivatives at the boundary c0 must satisfy transformation matrix. Following the notation of Van de dψ (c) dψ (c) Walle22 and Herring and Vogt,48 the absolute energy shift Eτ τ k = τ k wB,cc wW,cc . (11) of the τ-valley for a homogeneous deformation described by dc c=c− dc c=c+ 0 0 the strain tensor in the x-basis is given by The confinement potential V (c,k) can be expressed as τ = τ + τ τ τ ij E dδij uqi qj , (19) | | = 0ifc
125319-4 VALLEY DEGENERACY IN BIAXIALLY STRAINED ... PHYSICAL REVIEW B 84, 125319 (2011) of the matrix Cij vanish and C11 = C22 = C33,C12 = C13 = IV. ROBUST VALLEY DEGENERACY AND CROSSOVER C23, and C44 = C55 = C66, simplifying the stress tensor σ to VALLEY DEGENERACY ⎛ ⎞ ⎛ ⎞ C11 C12 C12 000⎛ ⎞ Two valleys τ and τ will be degenerate if σ1 ⎜ ⎟ ⎜ ⎟ 1 σ ⎜C12 C11 C12 000⎟ τ = = τ = ⎜ 2⎟ ⎜ ⎟ ⎜2⎟ E (k 0) E (k 0). (33) ⎜ ⎟ ⎜C C C 000⎟ ⎜ ⎟ ⎜σ3⎟ ⎜ 12 12 11 ⎟ ⎜3⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ , (26) When this condition holds for τ and τ regardless of the QW ⎜σ ⎟ ⎜ 000C44 00⎟ ⎜4⎟ ⎜ 4⎟ ⎜ ⎟ ⎝ ⎠ width, we call the valley degeneracy robust, and if it holds only ⎝ ⎠ 5 σ5 ⎝ 0000C44 0 ⎠ for a specific well width W0, we call it crossover degeneracy. 6 σ6 00000C44 The crossover-degeneracy condition can arise only when the valley electrons with a higher confinement mass are strain- where 1 = xx, 2 = yy, 3 = zz, 4 = 2yz, 5 = 2zx, and shifted to a higher energy than the valley electrons with a 6 = 2xy. Table I lists the values of C11,C12, and C44 for lower confinement mass. AlAs and GaAs. The strain tensor and the elastic constants Robust valley degeneracy can only occur if the strain ener- cklmn are used to obtain the free energy of isothermal elastic gies and out-of-plane confinement masses are independently deformations of a medium,28 equal. The strain energy must satisfy 1 Eτ = Eτ , (34) F ( ) = c . (27) 2 klmn kl mn klmn for all well widths, which simplifies via Eq. (19)tothex-basis For a cubic crystal, using the Voigt notation for the elastic condition constants, the elastic energy reduces to qτ · · qτ = qτ · · qτ (35) C = 11 2 + 2 + 2 + 2 + 2 + 2 or equivalently in the a-basis, F ( ) xx yy zz 2C44 xy xz yz 2 τ τ τ τ q · · q = q · · q . (36) + C12(xxyy + xxzz + yyzz). (28) M = M = When D1 D2 0, this condition for degeneracy simply M reduces to The coefficients Di can be determined for a given crystal τ τ facet orientation by transforming the strain tensor of into |q · c|=|q · c|, (37) the x-basis with the rotation matrix: meaning that all valleys with the same polar angle from the = (RM )−1( )RM , (29) c axis are degenerate. For the more general case of arbitrary M Di ,thefullEq.(36) has to be solved to determine valley and inserting these components into the elastic energy of degeneracy. However, given the shear vector α in the plane of M Eq. (28) and minimizing with respect to each Di : the QW as defined in Eq. (32), a special case can be defined, and two valleys τ and τ are degenerate if they simultaneously = M −1 M F ( ) F [(R ) ( )R ], (30) satisfy Eq. (37) and dF( ) = τ τ M 0, (31) |q · α|=|q · α|, (38) dDi or equivalently, if the valleys are mirror symmetric about the to deduce a set of three equations, which can be simultaneously c-α plane. solved to give the DM values. With the DM values solved for i i The second condition for robust valley degeneracy is that this material and this facet, the strain tensor in the x-basis can the out-of-plane inverse masses are equal, nowbeusedinEqs.(19)–(21) to deduce the strain-energy contribution to each valley Eτ . τ = τ w cc w cc, (39) As discussed in Sec. VII, for high-symmetry crystal orientations like (001), (110), and (111), there is no shear thus guaranteeing equal confinement energies. As long as the component in the strain tensor when expressed in the transport mass ellipsoid associated with each valley is oriented with its M = M = longitudinal mass parallel to the unit vector qτ ,asisusually a-basis so that D1 D2 0, whereas for (411) growth the M the case, then Eq. (39) is automatically satisfied under the same D2 coefficient is nonzero, and the direction of shear is defined by the vector α given by condition as Eq. (37).
α = M + M D1 a D2 b. (32) V. PIEZOELECTRIC EFFECTS ON THE QUANTUM-WELL M POTENTIAL For AlAs and GaAs, we tabulate the nonzero Di in Table I along with the elastic stiffness components in Voigt In crystals that are inversion asymmetric, like zinc blende notation. For completeness, the Appendix provides a compact crystals, strain may also produce piezoelectric fields, modify- list of equations for determining the strain tensor for arbitrary ing the confinement potential V (c,k) and resulting in shifts of τ substrate orientation. From the above analysis, we can now the ground energies E0 for the various valleys as solutions to calculate all the terms in Eq. (7) for any valley and any QW the Schroedinger¨ equation (10). The out-of-plane piezoelectric orientation. field, which will not break robust valley degeneracy, will shift
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M X TABLE I. Strain ratios Di for different facet orientations, elastic constants Cij , deformation potentials ac and u , X-valley mass tensor 21,22,52–54 components ml and mt in units of me, and piezoelectric coefficient ex,4 for AlAs and GaAs.
001 110 111 411 411 X 2 Material D0 D0 D0 D0 D2 C11,GPa C12,GPa C44,GPa ac,eV u ,eV ml mt ex,4,C/m AlAs 0.854 0.616 0.550 0.775 0.176 125.0 53.4 54.2 2.54 6.11 1.1 0.20 −0.23 GaAs 0.934 0.580 0.489 0.820 0.250 122.1 56.6 60.0 −0.16 8.61 1.3 0.23 −0.16
the crossover-degeneracy condition to a different well width, we obtain W . ⎛ ⎞ 0 1 In our treatment, we will calculate the piezoelectric field ⎛ ⎞ ⎛ ⎞ E ⎜2⎟ inside the QW and assume that any out-of-plane component x − 000ex,4 00⎜ ⎟ ⎜ ⎟ 1 ⎝ ⎠ ⎜3⎟ ⎝Ey ⎠ = 000 0 e 0 ⎜ ⎟ . will be canceled with an external gate bias, restoring the x,4 ⎜ ⎟ E εsε0 4 condition of the flat square well. The piezoelectric field inside z 000 0 0 ex,4 ⎝ ⎠ 5 the QW and/or the barriers is proportional to the shear strain 6 in the crystal basis27 (41) −1 Ei = ei,j j . (40) Table I lists the piezoelectric coefficients for GaAs and AlAs εsε0 and the coefficient for Al Ga − As can be linearly interpolated E x 1 x Here, i lists the unprimed piezoelectric-field components according to the x-Al content.27 The general shear strain = in the x-basis indexed by i x,y,z, ε0 and εs are the components in a QW are reproduced from Cadini and Stark24,55 free-space and relative semiconductor dielectric constants, and and simultaneously derived using Eqs. (29), (23), (30), and ei,j represents the piezoelectric tensor with j indices in Voigt (31). It is useful to express the electric field in the primed notation. We assume that Ei is zero in the unstrained substrate. − growth a-basis Recall that = (RM ) 1( )RM is the strain tensor derived from ⎛ ⎞ ⎛ ⎞ E Eq. (23) but expressed in the unprimed x-basis. For a zinc a − yz ⎜ ⎟ 2ex,4 M ⎝ ⎠ blende crystal structure, even though the facet orientations ⎝E b⎠ = R zx . (42) (110) and (111) have no shear strain in the growth a-basis, εsε0 E c xy they do have shear strain in the crystal x-basis. The (001) facet E orientation on the other hand does not have any shear strain. The c component of the piezoelectric field c will affect The strain in Eq. (40) is represented in Voigt notation as for the electron subband energy by modifying the confinement Eq. (26) with j = 1,2,...6. For zinc blende crystals, the only potential V (c,k)ofEq.(12) both in the strained well and in 27 nonzero piezoelectric coefficients are ex,4 = ey,5 = ez,6 and the strained barrier:
⎧ E | | ⎨⎪ q c,Wc, if c
E E where c,W and c,B are the out-of-plane c components of the VI. VALLEY-SCATTERING UNIT CELL electric field in the well and barrier, respectively. As pointed out by Adachi27 for (111) GaAs and AlAs, this piezoelectric In addition to calculating valley subband energies and field points from the (111)B surface (As-terminated) to the degeneracies, it is useful to visualize the valley orientations in (111)A surface (Ga-terminated) when under positive c axis three dimensions in order to map relevant intervalley scattering strain. vectors. In this section, we lay down a general derivation of a The piezoelectric field in the QW and barrier material alters 3D primitive unit cell, which we call the valley-scattering unit both the quantum confinement energy and the wave function, cell (VSC), that aids in identifying in-plane and out-of-plane and in the limit of weak piezoelectric fields, the energy shift intervalley scattering vectors. The proposed VSC is not a 3D τ = E Brillouin zone, but a simple unit cell defined to have the can be estimated to be Epz cW/2. For example, in strained (111) AlAs on a GaAs substrate, the piezoelectric field is 2D symmetry of the Miller index plane of the QW, while E = × 6 = preserving the total volume of the original 3D Brillouin zone. c 8.794 10 V/m, so for a QW width of W 10 nm the τ = piezoelectric energy shift would be around Epz 43.97 meV. This cell is to be also distinguished from the standard depiction For the present treatment, we reiterate our assumption that of a 2D Brillouin zone, which neglects the information about E any nonzero out-of-plane component of piezoelectric field c the out-of-plane momentum, which is necessary for calculating will be assumed to be canceled by an external gate voltage. momentum-scattering matrix elements.
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To define such a unit cell, we first identify the 2D subset Quantum confinement is created by sandwiching the AlAs of reciprocal Bravais lattice points that lie in the Miller index QWs between aluminum gallium arsenide (Alx Ga1−x As) lay- plane of the QW. The original 3D Bravais lattice is composed ers, which have a high aluminium content x>0.4. Properties of parallel planes of this 2D sublattice, which are displaced of the barrier alloy are determined by interpolating between and separated by an interplanar wave vector 2π/d.Wenow AlAs and GaAs, either linearly or with a bowing term where take the primitive unit cell of this 2D sublattice and extend it applicable. We assume the reciprocal mass tensor in the in the perpendicular direction by an amount ±π/d, resulting Alx Ga1−x As barrier layer to follow a linear interpolation for in a primitive unit cell with the same volume as the original useinEqs.(10) and (11),56 3D Brillouin zone. We call the resulting unit cell the VSC. wτ = (x)wτ + (1 − x)wτ . (46) We define coordinates for each valley τ so that it lies within Alx Ga(1−x)As AlAs GaAs the VSC. QW structures induce characteristically anisotropic τ 52 For Alx Ga1−x As, EB is given by the relation scattering potentials, defined in the a-basis as Vs(a,b,c), that can result in anisotropic intervalley scattering. Scatterers such Eτ = xEτ + (1 − x)Eτ − bx(1 − x). (47) Alx Ga1−x As AlAs GaAs as a single monolayer step in the sidewall of a QW or a miscut substrate with periodic sidewall steps are sharp on the order of 52 57 τ τ For Alx Ga1−x As, the bowing term b is 0.055 eV for EGaAs a single lattice period and could in principle induce large mo- τ defined to be zero and E = 0.259 eV, where τ can be Xx , mentum intervalley scattering in the k space plane parallel to AlAs Xy ,orXz. the QW. For all strain calculations, we assume AlAs to be strained ττ Recalling that the intervalley-scattering vector is Q ,we with respect to the GaAs substrate because in typical structures calculate the intervalley-scattering matrix element, the thickness of the intervening buffer Alx Ga1−x As layer is below the critical strain relaxation thickness of 0.5 μm.58,59 Using Eq. (22), we can calculate the in-plane strain from the v = ψ (r)|V (a,b,c)|ψ (r) (44) ττ τ s τ lattice mismatch between the AlAs QW and the GaAs substrate iQττ ·(a+b+c) = φ(r)u τ (r)|V (a,b,c)e |φ(r)u τ (r) q s q − = aGaAs aAlAs ∗ ∗ ττ · ττ · + , (48) = 3 iQ⊥ c iQ (a b) d rφ (r)uqτ (r)Vs(a,b,c)e e aAlAs = × φ(r)uqτ (r). (45) where aGaAs 0.564 177 nm (0.565 325 nm) is the lattice constant of the GaAs substrate and aAlAs = 0.565 252 nm (0.566 110 nm) is the lattice constant of the AlAs at 0 K To emphasize the importance of the out-of-plane intervalley (300 K).52 The perpendicular strain component is calculated ττ ⊥ scattering vector, we write Q in the exponential as the sum of using Eq. (23), where DM is the biaxial Poisson ratio in the ττ ττ 0 an out-of-plane component Q⊥ and the in-plane vector Q . respective growth direction. The strain energy is determined If we use the standard planar 2D Brillouin zone view, we lose from the absolute deformation potential for uniaxial strain at the representational importance of the out-of-plane scattering X = the X point, u 6.11 eV in AlAs. component. Therefore, we choose to depict the VSC in all three In the following subsections, we compare and contrast dimensions. Specific examples will be shown in the following the calculated results for the (001)-, (110)-, (111)-, and section, which demonstrate how all possible intervalley- (411)-oriented QWs, as summarized in Fig. 2.Intheleft scattering events can be easily identified in such a cell. panel of the figure, the transport basis vector c is taken along the growth direction. The differently colored ellipsoids distinguish the triple- (purple), double- (red), and single- VII. VALLEY DEGENERACY AND THE (blue) degenerate valleys, each of these satisfy the robust VALLEY-SCATTERING UNIT CELL FOR AlAs QWs degeneracy condition given by Eqs. (37)or(38). For the (001), (110), and (411) orientations, the strain energy breaks We apply the above analysis to the AlAs multivalley system, the threefold valley degeneracy into twofold valleys and a determining the valley vectors, mass tensors, and strain tensors single valley. The growth direction governs which of these for the various growth directions, and then we calculate the valleys will have the lowest energy for a given range of valley subband energies as a function of well width W for each well widths. For the (111) orientation, symmetry dictates orientation. The three degenerate conduction-band electron threefold-degenerate Xx,y,z valleys for all well widths. valleys are composed of six half valleys located at the X points The right panel of Fig. 2 shows the results of the analytical 100 = of the Brillouin zone edge, with valley vectors q (2π/a)x, simulation of Eτ , which gives the valley subband energies as a 010 = 001 = 0 q (2π/a)y, and q (2π/a)z for the Xx ,Xy , and Xz function of the well width. The horizontal dashed lines indicate points, respectively. The superscripts express τ as momentum- the pure strain splitting energy Eτ , equivalent to the energy 22 space directions after the notation of Van de Walle. The splitting in the asymptotic wide-well limit. The analytical AlAs electron valleys have anisotropic electron mass, with results are confirmed by the semiconductor heterostructure = 60,61 heavy longitudinal mass ml 1.1 me and light transverse simulation software NEXTNANO. For the crossover QW = 1,12,13 mass mt 0.20 me, respectively. In the conventional widths W0 indicated for the (001), (110), and (411) orienta- τ τ = crystal x-basis, the mass tensor m is diagonal with mii ml tions, all valleys satisfy the degeneracy condition of Eq. (33) τ τ = for the mass component parallel to q and mjj mt for the simultaneously. The intervalley-scattering momentum vectors two transverse mass components, respectively. Qττ for all the facets are listed in Table II.
125319-7 S. PRABHU-GAUNKAR et al. PHYSICAL REVIEW B 84, 125319 (2011)
FIG. 2. (Color online) Depiction of the valley degeneracies for various QW orientations. Each row is labeled with its Miller index. Left column: 3D representation of the X valleys showing both the CCC kx ,ky ,kz and the transport basis ka,kb,kc. In all diagrams, the double-degenerate Xx,y valleys are red and the singly degenerate Xz valley is blue. For the (111)-oriented AlAs QW, the three degenerate Xx,y,z valleys are purple. We note that the red and purple valleys exhibit robust valley-degeneracy condition whilst the intersection point of the red and blue traces in the rightmost column identifies the crossover-degeneracy condition. The purple valleys satisfy the robust valley-degeneracy condition as well as the crossover-degeneracy condition trivially for all well widths. Center column: 2D projection and 3D representation of the valley-scattering unit cell described in the text. Right column: Valley-degenerate ground energy as a function of QW width for a QW barrier of Al0.45Ga0.55As at T = 4 K. Dashed lines represent the calculated strain-energy shift of the respective electron valley relative to unstrained AlAs. The results of the heterostructure simulation software NEXTNANO are depicted with scatter plots and verified with the analytical calculations shown as continuous lines. The (111) and (411) calculations assume that the piezoelectric field has been canceled by a top gate to result in a flat QW.
A. (001) AlAs is given by22 DM = 2 c12 , and the diagonal strain matrix has 0 c11 001 = 001 = For the (001)-oriented QW, the crystal axes are identical to D1 D2 0. The strain tensor in the a-basis is the growth axes, Fig. 2 (top row), thus the coordinate transfor- 001 = ⎡ ⎤ mation matrix is the identity matrix R I. The inverse mass 10 0 tensor follows trivially. The various mass parameters listed in = ⎣01 0 ⎦ . (49) Table II are then used to calculate the respective energy terms 00−0.854 in Eq. (7). The biaxial Poisson ratio derived from Sec. III
125319-8 VALLEY DEGENERACY IN BIAXIALLY STRAINED ... PHYSICAL REVIEW B 84, 125319 (2011)
TABLE II. Reciprocal mass tensors wτ for the various valleys in the transport basis for (001)-, (110)-, (111)-, and (411)-oriented AlAs QWs. Intervalley-scattering vectors Qττ are listed, as are the 2D density-of-states (cyclotron) masses relative to the free-electron mass. The τ 2D density-of-states values n2D are combined for robust-degenerate valleys. M (001) τ [100] [010] [001] ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0.91 0 0 50 0 500 Q100,010 =( 1 −10)2π ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ a w τ = wτ ⎣ 050⎦ ⎣ 00.91 0 ⎦ ⎣ 050 ⎦ 010,001 2π Q =( 0 1 0 ) a 001,100 2π 005 00 5 000.91 Q =( 1 0 0 ) a τ m2D [me] 0.469 0.200 τ −2 −1 11 10 n2D [cm meV ]3.92× 10 8.36 × 10 M (110) τ [100] [010] [001] ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 50 0 50 0 0.91 0 0 Q100,010 =( 0 −1.4142 0 ) 2π ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ a w τ ⎣ 02.95 −2.05 ⎦ ⎣ 02.95 2.05 ⎦ ⎣ 050⎦ 010,001 2π Q =( 1 0.7071 −0.7071) a 001,100 2π 0 −2.05 2.95 02.05 2.95 005 Q =(−10.7071 0.7071) a τ m2D [me] 0.260 0.469 τ −2 −1 11 11 n2D [cm meV ]2.18× 10 1.96 × 10 M (111) τ [100] [010] [001] ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 2.95 −1.18 −1.67 2.95 1.18 1.67 50 0 Q100,010 =(1.2247 −2.1213 0) 2π ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ a w τ ⎣ −1.18 4.32 −0.96 ⎦ ⎣ 1.18 4.32 −0.96 ⎦ ⎣ 02.27 1.93 ⎦ 010,001 2π Q =(2.4494 0 0) a 001,100 2π −1.67 −0.96 3.64 1.67 −0.96 3.64 01.93 3.64 Q =(1.2247 −0.7071 0) a τ m2D [me] 0.296 τ −2 −1 11 n2D [cm meV ]3.72× 10 M (411) τ [100] [010] [001] ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 50 0 2.95 −1.93 −0.68 2.95 1.93 0.68 Q100,010 =(−0.7071 0.6667 0.2357) 2π ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ a w τ ⎣ 04.55 1.29 ⎦ ⎣ −1.93 3.18 −0.64 ⎦ ⎣ 1.93 3.18 −0.64 ⎦ 010,001 2π Q =(−1.4141 0 0 ) a 001,100 2π 01.29 1.36 −0.68 −0.64 4.77 0.68 −0.64 4.77 Q =( 0.7071 −0.6667 −0.2357) a τ m2D [me] 0.210 0.420 τ −2 −1 10 11 n2D [cm meV ]8.77× 10 3.52 × 10
The results of the Sec. III analysis for (001)-oriented QWs right-handed coordinate system. The coordinate transforma- including 2D density-of-states and cyclotron mass for each tion matrix R110 is valley are provided in Table II. ⎡ ⎤ The center panel of Fig. 2 (001) depicts the VSC and its 001 ⎢ − ⎥ 110 = ⎣ √1 √1 0⎦ 2D projection. The 2D sublattice of coplanar reciprocal lattice R 2 2 . (50) points is square symmetric, with the Xz valleys lying at the √1 √1 0 2 2 corners of the unit cell and the Xx,y valleys at the edges. The resulting VSC illustrates that all valleys can be connected by The biaxial Poisson ratio derived in Sec. III is given by22 + − purely in-plane scattering events, which is not obvious from DM = c11 3c12 2c44 , and the diagonal strain matrix has D110 = 0 c11+c12+2c44 1 the usual depiction of (001) valleys. This can be verified in 110 ττ D = 0. Table II where the c component of the scattering vector Q⊥ is 2 zero for all intervalley-scattering vectors. The center panel of Fig. 2 (110) depicts the VSC and its As seen in Fig. 2 (001, right), strain alone, as indicated 2D projection. The strain tensor is diagonal in the a-basis and by the horizontal dashed lines, contributes an energy shift given by ⎡ ⎤ of E100,010 =−11.62 meV for the doubly-degenerate X x,y 10 0 valleys while raising the singly degenerate strained X valley ⎢ ⎥ z = by E001 =+9.93 meV, a net difference of = 21.55 meV. ⎣01 0 ⎦ . (51) The ordering of the degeneracy changes at the crossover well 00−0.616 001 = width W0 5.4 nm. Because (001) biaxial strain induces no shear component in The results of Sec. III analysis for (110)-oriented QWs the crystal basis, there are no piezoelectric fields for this facet. including 2D density-of-states and cyclotron masses for each valley are provided in Table II. The 2D sublattice that defines the VSC has a centered- B. (110) AlAs rectangular symmetry. The Xz valley lies in this plane and For the (110)-oriented QW in Fig. 2 (110, left), we apply the Xx,y valleys lie outside of the central plane. In the VSC, the rules of axis identification described in Sec. III.Unit it is clear that whereas the Xx,y valleys can scatter amongst vector a lies along the lowest Miller index [001] direction, each other with purely in-plane scattering, the X valley is z making the Xz valley in-plane and b along [110] completes the isolated and requires an out-of-plane component for intervalley
125319-9 S. PRABHU-GAUNKAR et al. PHYSICAL REVIEW B 84, 125319 (2011) scattering. Table II reflects this result since only the Q100,010 valley are provided in Table II. The (111) QW valleys remain scattering vector has a zero c component. threefold degenerate due to equal contributions of the strain As seen in Fig. 2 (110, right), the singly degenerate Xz val- and confinement energies for all electron valleys as seen in ley has the lowest strain energy E001 =−12.94 meV whereas Fig. 2 (111) left. the doubly degenerate strained Xx,y valleys have an energy of The center panel of Fig. 2 (111) depicts the VSC and its E100,010 =−3.55 meV due to the smaller compressive strain. 2D projection. The planar sublattice is hexagonal, with six A smaller strain differential of = 10.77 meV is observed half-valleys located at the center of the hexagonal facets. between the valleys in this orientation as compared to the The resulting VSC shows that the valleys are all connected (001) orientation. We observe the valley-degeneracy crossover with coplanar scattering vectors, as seen in Table II from 110 = ττ at W0 5.3 nm. the zero c component of all Q . Figure 2 (111, right) A purely in-plane piezoelectric field arises in (110) QWs shows that all the valleys in this triply degenerate system are due to the nonzero shear component of the strain tensor in the equally strained to E100,010,001 =−7.0 meV and the robust x-basis. When Eq. (23) is transformed to the x-basis, the shear degeneracy condition given by Eq. (33) is trivially satisfied for components take the form55 all well widths.
−C11 − 2C12 A purely out-of-plane piezoelectric field arises in (111) = ||,= = 0, (52) xy C + C + 2C yz zx QWs due to the nonzero shear component of the strain tensor 11 12 44 in the x-basis. When Eq. (23) is transformed to the x-basis, the leading to an electric field in the QW plane parallel to the [100] shear components take the form24,27 a direction: − − − = = = C11 2C12 E = 2ex,4 xy yz zx ||, (56) a xy. (53) C11 + 2C12 + 4C44 εsε0 leading to an electric field perpendicular to the QW plane E = × 6 This electric field a 7.942 10 V/m plays no role in parallel to the c growth axis: quantum confinement so that the flat QW assumption remains √ − valid, and the field is screened in plane by the electrons in the E = 2ex,4 3 c xy. (57) QW within a Thomas-Fermi screening length of the sample εsε0 edges at ±a. E = × 6 We will assume that this electric field c 8.794 10 V/m will be canceled by a gate voltage, so that the flat QW C. (111) AlAs assumption remains valid. For the (111)-oriented QW, a is chosen along the lowest Miller index [110] direction perpendicular to the growth axis D. (411) AlAs c, and none of the valleys lie within the plane. Unit vector b is chosen along the [112] direction to complete the right- To demonstrate the utility of the formalism described in handed coordinate system.20 The coordinate transformation this paper, we now extend our analysis to the less conventional matrix R111 is (411) AlAs QW, which has nonetheless shown important ⎡ ⎤ results in the literature.6,62 Applying the same rules of axis √1 √−1 0 ⎢ 2 2 ⎥ identification, a is chosen along the [011] direction, which is 111 ⎢ √1 √1 √−2 ⎥ the lowest Miller index along the plane of the QW. Unit vector R = ⎣ ⎦ . (54) 6 6 6 b is chosen along the [122] direction to complete the right- √1 √1 √1 3 3 3 handed coordinate system. The coordinate transformation matrix R411 is The biaxial Poisson ratio derived in Sec. III is given by22 ⎡ ⎤ + − 1 −1 M = c11 2c12 2c44 0 √ √ D0 2 c +2c +4c , and the strain matrix is diagonal with 2 2 11 12 44 ⎢ − ⎥ D111 = D111 = 0. The strain tensor in the a-basis is R411 = ⎣ 1 2 2 ⎦ . (58) 1 2 ⎡ ⎤ 3 3 3 √4 √1 √1 10 0 18 18 18 = ⎣ ⎦ 01 0 . (55) Using the derivation for the strain ratios from Sec. III,for 00−0.55 411 = 411 = the (411)-oriented QWs, we obtain D0 0.775 and D2 The results of the Sec. III analysis for (111)-oriented QWs 0.176. The explicit relations for the biaxial Poisson ratios for including 2D density-of-states and cyclotron mass for each AlAs (411)-QWs are
6(c + 2c )(4c − 4c + 19c ) D411 = 11 12 11 12 44 − 1, 0 8c2 − (16c − c )(c + 2c ) + c (8c + 145c ) 11 12√ 44 12 44 11 12 44 −15 2(c + 2c )(c − c − 2c ) D411 = 11 12 11 12 44 . (59) 2 2 8c11 − (16c12 − c44)(c12 + 2c44) + c11(8c12 + 145c44)
125319-10 VALLEY DEGENERACY IN BIAXIALLY STRAINED ... PHYSICAL REVIEW B 84, 125319 (2011)
The results of Sec. III analysis for (411) including 2D From Fig. 2 (411, right), we obtain a crossover width 411 = density-of-states and cyclotron mass for each valley are W0 5.0 nm, and the valley occupation as a function provided in Table II. For this high-index facet, the strain tensor of well width follows an analogous discussion as for the is nondiagonal in the a-basis and given by (110) case. However, we observe that the Xx valley is ⎡ ⎤ singly degenerate and the X valleys are doubly degen- 10 0 y,z ⎣ ⎦ erate. The strain component of the Xx valley energy is = 01 0.176 . (60) 100 E = 6.82 meV, and that of the Xy,z valleys, which 00.176 −0.775 satisfy the robust valley-degeneracy condition, is E010 = − = The biaxial strain in the structure grown along [411] gives 12.29 meV. A strain differential of 19.11 meV is ob- tained. The two degenerate valleys have coplanar momentum rise to nondiagonal components in the strain tensor. The effect 010,001 of shear is depicted in Fig. 3 by the shear vector α [see scattering as seen in the vector Q , which has zero Eq. (32)], which lies in the plane of the QW and is parallel to c component. the b direction. The degeneracy condition takes into account A piezoelectric field with both in-plane and out-of-plane 411 components arises in (411) QWs due to the nonzero shear the shear component D1 and is given by Eq. (38). Figure 2 (411, center) depicts the VSC and its 2D projection. component of the strain tensor in the x-basis. When Eq. (23) is transformed to the x-basis, the shear components take the The 2D planar sublattice in Fig. 2 (411, center) has centered 24 rectangular symmetry. form
−(C11 + 2C12)(17C11 − 17C12 + 2C44) = = ||, xy zx 2 − − + + + 8C11 (16C12 C44)(C12 2C44) C11(8C12 145C44) −(C11 + 2C12)(8C11 − 8C12 − 7C44) = ||, (61) yz 2 − − + + + 8C11 (16C12 C44)(C12 2C44) C11(8C12 145C44)
leading to an electric field with components both in the QW VIII. THE VALLEY-SCATTERING UNIT CELL AND ab plane as well as parallel to the c growth axis: ANISOTROPIC INTERVALLEY SCATTERING ⎛ ⎞ FOR AlAs QWs − √1 + √1 2 xy 2 zx 2ex,4 ⎜ 4 1 ⎟ This section first discusses the differences between the VSC E =− ⎝ + xy − yz ⎠ . (62) ε ε 3 3 and standard depiction of the 2D Brillouin zone, and then s 0 √2 + √4 18 xy 18 yz determines the inter to intravalley scattering ratio for valleys near crossover degeneracy. In the VSC description for the case We will assume that the out-of-plane piezoelectric-field of (111)AlAs, Fig. 4 shows the (111) AlAs VSC as well as E = × 6 component c 1.783 10 V/m will be canceled by a gate the 2D Bravais lattice of these cells. For comparison with this voltage so that the square-QW assumption remains valid. The hexagonal unit cell, the standard 2D Brillouin zone is plotted as E 21 piezoelectric-field component a is zero, and the in-plane com- a smaller grey hexagon within this lattice. The hexagons A, E = × 6 ponent b 3.761 10 V/m will be screened at the sample B, and C are intended to illustrate the relative positions of three edges by the electrons in the QW.
FIG. 4. (Color online) 3D representation of the valley-scattering cell for (111)AlAs as well as 2D lattice of such cells. In AlAs, the valleys are located at the q = (2π/a)( x, y, z). The standard depiction of 2D Brillouin zone (small shaded hexagon)21 projects together FIG. 3. (Color online) Illustration of shear strain for the (411)- three identical but laterally and vertically displaced planar hexagonal 411 oriented QWs due to the nonzero strain-to-shear ratio D1 .Pa- lattices, translated according to the representative hexagonal cells rameters a, b,and c represent strain displacements along the A, B, and C. The area of standard 2D Brillouin zone is one third respective vectors of the transport a-basis, and α represents the shear that of the valley-scattering cell since its height is three times that vector in the plane of the QW, which for the (411)-oriented QWs is of the valley-scattering cell. Arrows illustrate the planar intervalley- parallel to the b axis. scattering vectors Qττ for (111)AlAs.
125319-11 S. PRABHU-GAUNKAR et al. PHYSICAL REVIEW B 84, 125319 (2011) identical planar hexagonal lattices, displaced from each other. a suppression of more than 10 orders of magnitude for interval- When these three layers are stacked in sequence ABCABC- out ley scattering. Thus intervalley surface-roughness scattering is of the plane we recover the full 3D reciprocal lattice. Whereas significantly suppressed with respect to intravalley scattering. the standard 2D Brillouin zone is defined from the overlay Alloy scattering is the other possible mechanism for of all three layers, we define the VSC from the 2D Brillouin elastic intervalley scattering. The standard model for alloy zone of a single layer only. The area of the hexagonal VSC for disorder scattering65,67 with variable alloy composition x(z)is (111) is thus three times larger than that of the traditional described by the autocorrelation function hexagonal unit cell. The coplanar intervalley momentum a3V 2 ∞ scattering vectors are shown with arrows in the VSC. ττ 2 = 0 0 2 − 2 UAD ψτ (c)x(c)[1 x(c)]ψτ (c)dc, (66) 8 W The VSC is helpful in depicting intervalley-scattering 2 events, so we will discuss the two main sources of single- where a0 is the in-plane lattice constant and V0 is the spatial particle scattering for AlAs QWs, namely interface-roughness average of the fluctuating alloy potential over the alloy unit and alloy-disorder scattering in the barrier. We find below cell. In AlAs QWs, only tails of the wave function have an that near valley degeneracy, interface-roughness scattering alloy content in the Alx Ga1−x As barrier. One notes that the does not result in any significant intervalley scattering, and wave functions of electrons in valleys with a light confinement in the presence of alloy scattering in the barrier walls, mass penetrate to a greater extent in the barrier and will have the scattering rate for crossover intervalley scattering is a significantly larger alloy-scattering potential as compared significantly repressed relative to robust intervalley scattering, to the heavy confinement mass. Assuming valley degeneracy and this suppression factor is calculated. for all valleys and plane-wave solution for the electron wave We assume low enough temperatures such that acoustic function in the QW barrier, the scattering matrix element can and optical phonons can be neglected. Elastic intervalley be written as scattering can only occur if the valley-splitting energy is less 3 2 ∞ τ τ a V τ τ than the Fermi energy at low temperatures |E − E | 125319-12 VALLEY DEGENERACY IN BIAXIALLY STRAINED ... PHYSICAL REVIEW B 84, 125319 (2011) (111) GaAs/AlAs growth has been shown to have superior model is the definition of two relevant bases: the conventional morphology with a miscut angle from 0.5◦ to 4◦ and, in crystal x and the transport a-basis. We use this notation to find general, intentional miscuts improve growth quality due to the projected in-plane and out-of-plane effective masses, the slip-step growth.69–72 In Si, recently investigated hydrogen- electron subband energy and degeneracy, piezoelectric fields, terminated (111) Si miscut surfaces have shown very high and the scattering vectors. mobility73 and the wafer miscut is expected to break the valley There are five competing energy scales in multivalley QWs degeneracy.43,74 namely in-plane kinetic energy, confinement energy, strain Miscut samples are characterized by two angles. One angle energy, piezoelectric energy, and Fermi energy, all of which φ designates the azimuthal angle in the ab plane relative to influence valley occupancy. Strain in the QW breaks valley the a axis toward which the plane is tilted, often expressed as degeneracy and results in an energy differential, Eτ of an in-plane Miller index tilt direction T . The other angle θ 0–20 meV between nondegenerate valleys. The strain energy is designates the polar tilt angle relative to the surface normal c. independent of the QW width unlike the quantum confinement Experimentally, the angles can be deduced from atomic-force energy, which increases as the QW width decreases and is microscope images of surface monolayer steps, where φ is inversely proportional to the confinement mass; typical con- = τ oriented perpendicular to the steps, and θ atan(a/2L)is finement energies E0 are of the order of 20 meV for a narrow deduced from knowing the monolayer thickness a/2 and the QW about 5-nm wide and 5 meV for a 20-nm-wide QW. average width of the monolayer steps L. We define a new We defined a robust valley-degeneracy condition to identify M coordinate transformation Rθ,φ for the miscut samples by valleys that are degenerate independent of well width, and a introducing the azimuthal and polar rotations Rφ and Rθ , crossover-degeneracy condition where valleys are degenerate respectively: only for a particular crossover well width W0. We showed that ⎡ ⎤ piezoelectric fields in the QW and barrier materials can arise in cosφ −sinφ 0 ⎢ ⎥ certain facets such as (111) and (411), and cause piezoelectric R = R = ⎣sinφ cosφ 0⎦ , (69) τ T φ energy shifts Epz as large as 50 meV for a (111) QW about 001 10-nm wide, and that such a piezoelectric field can be canceled ⎡ ⎤ by an external gate voltage. The smallest energy scale in these cosθ 0sinθ ⎢ ⎥ QWs is the Fermi energy, which is a few meV depending on the = 11 −2 Rθ ⎣ 010⎦ . (70) degeneracy for electron densities around n2D ∼ 3 × 10 cm τ ∼ −sinθ 0 cosθ and assuming an effective mass m2D 0.3me. With such a small Fermi energy, valley occupation is strongly defined by M We obtain a new coordinate transformation matrix Rθ,φ the interplay of the various other energy scales in the problem. for calculating the transport parameters and the strain tensor We once again emphasize that it is due to the similarity analysis of miscut samples of all the energy scales that none of them can be ignored − and must be treated carefully to calculate valley subband RM = (R R R 1)RM . (71) θ,φ φ θ φ energy. Note that a VSC cannot be defined for miscut samples, but Valley degenerate systems have an extra scattering channel only for those with a pure Miller index. not present in single-valley systems, namely, intervalley We outline here the procedure for determining valley scattering. We identified the main intervalley scattering mech- subband energies under miscut. First, the transformation anisms as the interface-roughness and alloy-disorder scattering M matrix Rθ,φ from Eq. (71) determines the inverse mass tensor due to wave-function penetration into the barrier and calculated of each valley with Eq. (9), and the out-of-plane inverse mass the suppression factor at the crossover well width W0 for tensor component can be used in Eq. (10) to determine the intervalley scattering between different sets of valleys. We also ground confinement energies of each valley. The in-plane defined a valley-scattering primitive unit cell to easily identify inverse mass tensor is then calculated with Eq. (14) and scattering events between the robust-degenerate and crossover- used in Eq. (15) to obtain the kinetic energy. The stress degenerate electron valleys. We drew the VSC for the special M tensor can be determined by applying Rθ,φ in Eq. (29), case of AlAs grown along four different orientations, the M high-symmetry facets (001), (110), and (111), as well as a and the strain ratios Di can be determined by minimizing the elastic energy with Eqs. (28)–(30). Finally, the strain- low-symmetry facet (411) to demonstrate the utility of our energy shift can be deduced from Eq. (19) and added to the model. Furthermore, we explained the relevance of the VSC ground quantum confinement and kinetic energies to arrive description for visual identification of anisotropic intervalley at the final valley energies. Any piezoelectric fields can be scattering in AlAs QWs. determined by transforming the strain tensor to the unprimed In the final section, we demonstrated the power of our for- frame and inserting the appropriate shear components into malism to determine valley degeneracy for arbitrary substrate Eq. (42). orientation, without being restricted to Miller-indexed planes as is prevalent in the literature. We consider the example of a miscut sample and define the additional notation required X. CONCLUSION to address angle deviation from conventional growth axes. In conclusion, we review the advantages and limitations of We then detail the procedure to calculate all the ground- the formalism we have developed for valley subband energy energy parameters as well as the strain-tensor analysis for calculations. We first derived how the valley index is a valid miscut samples. It is worth repeating that the formalism pseudospin index in a multivalley system. A key element of our developed here is sufficient to calculate valley degeneracy for 125319-13 S. PRABHU-GAUNKAR et al. PHYSICAL REVIEW B 84, 125319 (2011) any substrate orientation, and need not be aligned with any cij kl in the CCC x-basis, and the in-plane strain || in the Miller-index facet. transport a-basis. There are some important limitations of our formalism. We first obtain the fourth-order rotated elastic stiffness Firstly, we assume layer thicknesses are all within the strain- tensor cmnop in the a-basis by rotating the fourth-rank elastic relaxation limit such that there are no strain-induced defects stiffness tensor cij kl from the CCC basis x = (x,y,z) to the below or within the QW. These would relax the lattice transport basis a = (a,b,c), constant relative to the substrate, thus one would have to first = M M M M determine the adjusted lattice constant at the QW layer and cmnop Rmi Rnj RokRplcij kl , (A1) then apply the formalism developed here to deduce valley ij kl degeneracy. Secondly, we assume nearly empty QWs and do where RM is the coordinate transformation matrix defined not calculate the Schrodinger¨ equation self-consistently with in Eq. (8). The rotated fourth-rank tensor cmnop in the CCC the Poisson equation. However, our formalism for calculating x-basis should then be mapped to Cij using the Voigt notation. the various valley subband energies can be applied to standard The strain tensor components in the transport a-basis are self-consistent solvers to obtain the proper Hartree solution. ij given by Lastly, whereas the formalism takes into account the change a − a in the confinement potential due to piezoelectric fields, it does = = = substrate layer aa bb , (A2) not address larger structural considerations such as modulation alayer doping layers, surface pinning potentials, electrostatic charges, = = 0, (A3) and piezoelectric fields in the barrier, all of which would ab ba λμ − ηω have to be calculated together to determine the confinement = = = M ac ca D1 , (A4) well potential and Fermi energy. We note, however, that the λκ − η2 piezoelectric equations provided here can be used to determine ω − ηDM = = 1 = DM , (A5) the piezoelectric fields in the strained barriers. bc cb λ 2 α − 2C DM − 2C DM = = 34 2 35 1 =− M cc ⊥ D0 , (A6) ACKNOWLEDGMENTS C33 This work was funded by the NSF Grant DMR-0748856, where the denominators in Eqs. (A4), (A5) and (A7) are always the MRSEC program of the NSF DMR-0520513 at the nonzero, the coefficients α, β, and γ are first order, and the coefficients λ, κ, η, ω, and μ are second order in the elastic Materials Research Center of Northwestern University, and the BMBF Nano-Quit Project 01BM470. S.P.G. and M.G. would stiffness tensor components Cij : like to thank Wade DeGottardi, Wang Zhou, Florian Herzog, and Jens Koch for helpful discussions. =− + α (C13 C23), (A7) =− + β (C14 C24), (A8) APPENDIX: ANALYTICAL EQUATIONS OF THE STRAIN =− + γ (C15 C25), (A9) TENSOR AND THE STRAIN RATIOS FOR ARBITRARY = SUBSTRATE ORIENTATIONS λ 2C33C44, (A10) = M κ 2C33C55, (A11) In Sec. III, we derive the strain ratios Di by minimizing = − the free energy of the strained layer. This section presents η 2(C33C45 C34C35), (A12) a generalized treatment to express these strain ratios for an ω = C β − C α, (A13) arbitrary substrate orientation with cubic symmetry, given its 33 34 M = − coordinate transformation matrix R , the elastic constants μ C33γ C35α. (A14) 1M. Shayegan, E. P. De Poortere, O. Gunawan, Y. P. Shkolnikov, 6E. P. De Poortere, E. Tutuc, S. J. Papadakis, and M. Shayegan, E. Tutuc, and K. Vakili, Phys. Status Solidi B 243, 3629 Science 290, 1546 (2000). (2006). 7K. Vakili, Y.P.Shkolnikov, E. Tutuc, N. C. Bishop, E. P.De Poortere, 2T. P. Smith, W. I. Wang, F. F. Fang, and L. L. Chang, Phys. Rev. B and M. Shayegan, Physica E 34, 89 (2006). 35, 9349 (1987). 8H. W. vanKesteren, E. C. Cosman, P. Dawson, K. J. Moore, and 3T. S. Lay, J. J. Heremans, Y. W. Suen, M. B. Santos, K. Hirakawa, C. T. Foxon, Phys.Rev.B39, 13426 (1989). M. Shayegan, and A. Zrenner, Appl. Phys. Lett. 62, 3120 (1993). 9A. F. W. van de Stadt, P. M. Koenraad, J. A. A. J. Perenboom, and 4S. Dasgupta, C. Knaak, J. Moser, M. Bichler, S. F. Roth, A. F. I. J. H. Wolter, Surf. Sci. 362, 521 (1996). Morral, G. Abstreiter, and M. Grayson, Appl. Phys. Lett. 91,14 10O. Gunawan, Y. P. Shkolnikov, K. Vakili, T. Gokmen, E. P. De (2007). Poortere, and M. Shayegan, Phys. Rev. Lett. 97, 186404 (2006). 5S. Dasgupta, S. Birner, C. Knaak, M. Bichler, A. F. I. Morral, 11K. Vakili, Y. P. Shkolnikov, E. Tutuc, E. P. De Poortere, G. Abstreiter, and M. Grayson, Appl. Phys. Lett. 93,13 M. Padmanabhan, and M. Shayegan, Appl. Phys. Lett. 89,17 (2008). (2006). 125319-14 VALLEY DEGENERACY IN BIAXIALLY STRAINED ... PHYSICAL REVIEW B 84, 125319 (2011) 12O. Gunawan, Y. P. Shkolnikov, E. P. De Poortere, E. Tutuc, and 42C. Euaruksakul, F. Chen, B. Tanto, C. S. Ritz, D. M. Paskiewicz, M. Shayegan, Phys. Rev. Lett. 93, 246603 (2004). F. J. Himpsel, D. E. Savage, Z. Liu, Yugui Yao, Feng Liu, and 13O. Gunawan, E. P. De Poortere, and M. Shayegan, Phys. Rev. B 75, M. G. Lagally, Phys. Rev. B 80, 115323 (2009). 081304(R) (2007). 43K. Eng, R. N. McFarland, and B. E. Kane, Phys.Rev.Lett.99, 14Y. P. Shkolnikov, K. Vakili, E. P. De Poortere, and M. Shayegan, 016801 (2007). Phys.Rev.Lett.92, 246804 (2004). 44M. Eto and Y. Hada, AIP Conf. Proc. 850, 1382 (2006). 15J. Moser, T. Zibold, D. Schuh, M. Bichler, F. Ertl, G. Abstreiter, 45Y. Hada and M. Eto, Jpn. J. Appl. Phys. 43, 10 (2004). M. Grayson, S. Roddaro, and V. Pellegrini, Appl. Phys. Lett. 87,5 46Y. Hada and M. Eto, Phys.Rev.B68, 155322 (2003). (2005). 47J. Davies, in The Physics of Low-Dimensional Semiconductors 16J. Moser, S. Roddaro, D. Schuh, M. Bichler, V. Pellegrini, and (Cambridge University Press, Cambridge, UK, 2006). M. Grayson, Phys.Rev.B74, 193307 (2006). 48C. Herring and E. Vogt, Phys. Rev. 101, 944 (1956). 17O. Gunawan, B. Habib, E. P. De Poortere, and M. Shayegan, Phys. 49M. P. C. M. Krijn, Semicond. Sci. Technol. 6, 27 (1991). Rev. B 74, 155436 (2006). 50J. F. Nye, in Physical Properties of Crystals, Their Representation 18T. Gokmen, M. Padmanabhan, and M. Shayegan, Nat. Phys. 6, 621 by Tensors and Matrices (Oxford University Press, USA, 1957). (2010). 51W. Voigt, in Lehrbuch der Kristallphysik (Johnson Reprint Corpo- 19Y. P. Shkolnikov, S. Misra, N. C. Bishop, E. P. De Poortere, and ration, New York, 1966). M. Shayegan, Phys. Rev. Lett. 95, 066809 (2005). 52I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, J. Appl. Phys. 20M. Rasolt, in Advances in Research and Applications, Solid State 89, 5815 (2001). Physics Vol. 43, edited by H. Ehrenreich (Academic Press, San 53Halbleiter, Landolt-Boernstein, Vol. 17 (Springer, 1982). Diego, CA, 1990), pp. 93–228. 54S. H. Wei and A. Zunger, Appl. Phys. Lett. 72, 2011 (1998). 21F. Stern and W. E. Howard, Phys. Rev. 163, 816 (1967). 55Caridi and Stark incorrectly derived the equation for the strain- 22C. G. Van de Walle, Phys. Rev. B 39, 1871 (1989). tensor components of the (110) orientation. Following the appendix 23D. L. Smith and C. Mailhiot, J. Appl. Phys. 63, 8 (1988). of the present paper, the correct terms can be derived. 24E. A. Caridi and J. B. Stark, Appl. Phys. Lett. 60, 12 (1992). 56Adachi et al. found that in order to compare theory and experiment, 25L. De Caro, and L. Tapfer, Phys. Rev. B 48, 2298 (1993). interpolating the reciprocal mass followed by inverting it gives 26K. Yang, T. Anan, and L. J. Schowalter, Appl. Phys. Lett. 65,22 an equivalent result to inverting the reciprocal mass followed by (1994). interpolating it even though these two operations are mathematically 27S. Adachi, in GaAs and Related Materials: Bulk Semiconducting different. and Superlattice Properties (World Scientific Publishing, Singa- 57We have assumed that the bowing term from the expression for the pore, 1994), pp. 122–129. band gap in Ref. 52 is accounted for entirely in the conduction band. 28T. Hammerschmidt, P. Kratzer, and M. Scheffler, Phys.Rev.B75, 58M. Gutierrez, M. Herrera, D. Gonzalez, G. Aragon, J. J. Sanchez, 235328 (2007). I. Izpura, M. Hopkinson, and R. Garcia, Microelectron. J. 33, 553 29F. Schaffler,¨ Semicond. Sci. Technol. 12, 1515 (1997). (2002). 30R. D. Graft, D. J. Lohrmann, G. P. Parravicini, and L. Resca, Phys. 59C. Weisbuch and B. Winter, in Quantum Well Structures. Funda- Rev. B 36, 4782 (1987). mentals and Applications (Academic Press, New York, 1991). 31T. B. Boykin, G. Klimeck, M. Friesen, S. N. Coppersmith, 60C. Knaak, Diplomarbeit, Technische Universitat¨ Munchen,¨ 2007. P. vonAllmen, F. Oyafuso, and S. Lee, Phys.Rev.B70, 165325 61[www.nextnano.de]. (2004). 62E. D. Poortere, PhD. thesis, Princeton University, 2003. 32T. B. Boykin, G. Klimeck, M. A. Eriksson, M. Friesen, S. N. 63H. Sakaki, T. Noda, K. Hirakawa, M. Tanaka, and T. Matsusue, Coppersmith, P. von Allmen, F. Oyafuso, and S. Lee, Appl. Phys. Appl. Phys. Lett. 51, 23 (1987). Lett. 84, 1 (2004). 64T. Ando, A. B Fowler, and F. Stern, Rev. Mod. Phys. 54, 2 (1982). 33M. O. Nestoklon, L. E. Golub, and E. L. Ivchenko, Phys.Rev.B 65D. N. Quang, N. H. Tung, D. T. Hien, and H. A. Huy, Phys.Rev.B 73, 235334 (2006). 75, 073305 (2007). 34M. Friesen, M. A. Eriksson, and S. N. Coppersmith, Appl. Phys. 66D. N. Quang, V. N. Tuoc, and T. D. Huan, Phys.Rev.B68, 195316 Lett. 89, 202106 (2006). (2003). 35M. Virgilio and G. Grosso, J. Appl. Phys. 100, 9 (2006). 67F. Murphy-Armando, and S. Fahy, Phys. Rev. Lett. 97, 096606 36M. Friesen, S. Chutia, C. Tahan, and S. N. Coppersmith, Phys. Rev. (2006). B 75, 115318 (2007). 68W. Degottardi (private communication). 37S. Goswami, M. Friesen, L. M. McGuire, J. L. Truitt, C. Tahan, L. J. 69L. Vina and W. I. Wang, Appl. Phys. Lett. 48, 1 (1986). Klein, J. O. Chu, P. M. Mooney, D. W. van der Weide, R. Joynt, 70T. Hayakawa, M. Kondo, T. Suyama, K. Takahashi, S. Yamamoto, S. N. Coppersmith, and M. A. Eriksson, Nat. Phys. 3,1 and T. Hijikata, Jpn. J. Appl. Phys. 26, L302 (1987). (2007). 71A. Chin, P. Martin, P. Ho, J. Ballingall, T. H. Yu, and J. Mazurowski, 38M. O. Nestoklon, E. L. Ivchenko, J. M. Jancu, and P. Voisin, Phys. Appl. Phys. Lett. 59, 15 (1990). Rev. B 77, 155328 (2008). 72K. Tsutsui, H. Mizukami, O. Ishiyama, S. Nakamura, and 39M. Virgilio and G. Grosso, Phys.Rev.B79, 165310 (2009). S. Furukawa, Jpn. J. Appl. Phys. 29 (1990). 40G. Frucci, L. Di Gaspare, F. Evangelisti, E. Giovine, 73K. Eng, R. N. McFarland, and B. E. Kane, Appl. Phys. Lett. 87,5 A. Notargiacomo, V. Piazza, and F. Beltram, Phys. Rev. B 81, (2005). 195311 (2010). 74N. Kharche, S. Kim, T. B. Boykin, and G. Klimeck, Appl. Phys. 41J. Kim and M. V. Fischetti, J. Appl. Phys. 108, 1 (2010). Lett. 94, 4 (2009). 125319-15